Nilpotency degree of the nilradical of a solvable Lie algebra on two generators and uniserial modules associated to free nilpotent Lie algebras

Given a sequence (d) over right arrow = (d(1), ..., d(k)) of natural numbers, we consider the Lie subalgebra h of gl(d, F), where d = d(1)+ ... + d(k) and F is a field of characteristic 0, generated by two block upper triangular matrices D and E partitioned according to (d) over right arrow, and study the problem of computing the nilpotency degree m of the nilradical no f h. We obtain a complete answer when D and E belong to a certain family of matrices that arises naturally when attempting to classify the indecomposable modules of certain solvable Lie algebras. Our determination of m depends in an essential manner on the symmetry of E with respect to an outer automorphism of sl(d). The proof that mdepends solely on this symmetry is long and delicate. As a direct application of our investigations on hand n we give a full classification of all uniserial modules of an extension of the free l-step nilpotent Lie algebra on n generators when F is algebraically closed. (C) 2021 Elsevier Inc. All rights reserved.